Deck 6: Point Estimation

Question 1

Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu$ ?

A) The sample mean $\bar { X }$
Is always an unbiased estimator of $\mu$
)
B) The sample mean $\tilde { \mu }$
Is an unbiased estimator of $\mu$
If the distribution is continuous and symmetric.
C) Any trimmed mean is an unbiased estimator of $\mu$
If the distribution is continuous and symmetric.
D) None of the above statements are true.
E) All of the above statements are true.

Answer 1

All of the above statements are true.

Question 2

The sample median and any trimmed mean are unbiased estimators of the population mean if the random sample from a population that is __________ and __________.

Answer 2

continuous, symmetric

Question 3

Given four observed values: would result in a point estimate for that is equal to __________.

Answer 3

5

Question 4

Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu \text { and variance } \sigma ^ { 2 }$ ?

A) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 2 ) \text { is an unbiased estimator of } \sigma ^ { 2 }$
B) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 1 ) \text { is an unbiased estimator of } \sigma ^ { 2 }$
C) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / n \text { is an unbiased estimator of } \sigma ^ { 2 }$
D) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n - 1 ) \text { is an unbiased estimator of } σ^ { 2 }$

E) All of the above statements are true provided that the sample size n > 30.

Question 5

A point estimator is said to be an __________ estimator of if for every possible value of .

Question 6

In your text, two important methods were discussed for obtaining point estimates: the method of __________ and the method of __________.

Question 7

The standard error of an estimator is the __________ of .

Question 8

Which of the following statements are not always true?

A) A point estimator $\hat { \theta }$
Is said to be an unbiased estimator of parameter $\theta$
If $E ( \hat { \theta } ) = \theta$
For every possible value of $\theta$
)
B) If the estimator $\hat { \theta }$
Is not unbiased of parameter $\theta$
, the difference $E ( \hat { \theta } ) - \theta$
Is called the bias of $\hat { \theta }$
)
C) A point estimator $\hat { \theta }$
Is unbiased if its probability sampling distribution is always "centered" at the true value of the parameter $\theta$
, where "centered" here means that the median of the distribution of $\hat { \theta } \text { is } \theta$
)
D) All of the above statements are true.

Question 9

The objective of __________ is to select a single number such as , based on sample data, that represents a sensible value (good guess) for the true value of the population parameter, such as .

Question 10

Which of the following statements are not always true?

A) It is necessary to know the true value of the parameter $\theta$
To determine whether the estimator $\hat { \theta }$
Is unbiased.
B) When X is a binomial random variable with parameters n and p, the sample proportion $\hat { p } = X / n$
Is an unbiased estimator of p.
C) When choosing among several different estimators of parameter $\theta$
, select one that is unbiased.
D) All of the above statements are not always true.

Question 11

Which of the following statements are not true?

A) Maximum likelihood estimators are generally preferable to moment estimators because of certain efficiency properties.
B) Maximum likelihood estimators often require significantly more computation than do moment estimators.
C) The definition of unbiasedness in general indicates how unbiased estimators can be derived.
D) None of the above statements are true.
E) All of the above statements are true

Question 12

Which of the following statements are not true?

A) Maximizing the likelihood function gives the parameter values for which the observed sample is most likely to have been generated---that is, the parameter values that "agree most likely" with the observed data.
B) Different principles of estimation may yield different estimators of the unknown parameters.
C) The maximum likelihood estimator of the population standard deviation $\sigma$
Is the sample standard deviation S.
D) None of the above statements are true.

Question 13

Which of the following statements are true?

A) A point estimate of a population parameter $\theta$
Is a single number that can be regarded as a sensible value of $\theta$
)
B) A point estimate of a population parameter $\theta$
Is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of $\theta$
)
C) The sample mean $\bar { X }$
Is a point estimator of the population mean $\mu$
)
D) The sample variance $S ^ { 2 }$
Is a point estimator of the population variance $\sigma ^ { 2 }$
)
E) All of the above statements are true.

Question 14

Let be a random sample from a probability mass function or probability density function f(x). For k = 1,2,3,……, the kth population moment is denoted by __________, while the kth sample moment is __________.

Question 15

Let be a random sample of size n from an exponential distribution with parameter . The moment estimator of = __________.

Question 16

Which of the following statements are correct?

A) The first population moment is $\mu$
, while the first sample moment is $\bar { X }$
)
B) The moment estimators $\hat { \theta } _ { 1 } , \cdots \cdots , \hat { \theta } _ { m }$
Are obtained by equating the first m sample moments to the corresponding first m population moments, and solving for the unknown parameters $\theta _ { 1 } , \cdots \cdots , \theta _ { m }$
)
C) The method of maximum likelihood was first introduced by R.A. Fisher, a geneticist and statistician, in the 1920's.
D) All of the above statements are true.
E) Only (A) and (B) are true.

Question 17

Among all estimators of parameter that are unbiased, choose the one that has minimum variance. The resulting is called the __________ of .

Question 18

Which of the following statements are not true?

A) The symbol $\hat { \theta }$
Is customarily used to denote the estimator of parameter $\theta$
And the point estimate resulting from a given sample.
B) The equality $\hat { \mu } = \bar { X }$
Is read as "the point estimator of $\bar { X } \text { is } \hat { \mu } . "$
C) The difference between $\hat { \theta }$
And the parameter $\theta$
Is referred to as error of estimation.
D) None of the above statements is true.

Question 19

Let be the maximum likelihood estimates (mle's) of the parameters . Then the mle of any function h( ) of these parameters is the function of the mle's. This result is known as the __________ principle.

Question 20

An estimator that has the properties of __________ and __________ will often be regarded as an accurate estimator.

Question 21

A random sample of
a. Derive the maximum likelihood estimator of
. If
= 25 and
=5, what is the estimate?
b. Is the estimator of part (a) unbiased?
c. If
= 25 and
=5, what is the mle of the probability
that none of the next five helmets examined is flawed?

Question 22

Let represent a random sample from a Rayleigh distribution with pdf
a. It can be shown that
Use this fact to construct an unbiased estimator of
based on
(and use rules of expected value to show that it is unbiased).
b. Estimate
from the following
observations on vibratory stress of a turbine blade under specified conditions:
17.08 10.43 4.79 6.86 13.88
14.43 20.07 9.60 6.71 11.15

Question 23

Consider a random sample from the pdf where (this distribution arises in particle physics). Show that is an unbiased estimator of [ Hint: First determine

Question 24

Let denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of is where > -1. A random sample of ten students yields data
a. Use the method of moments to obtain an estimator of
and then compute the estimate for this data.
b. Obtain the maximum likelihood estimator of
and then compute the estimate for the given data.

Question 25

The accompanying data describe flexural strength (Mpa) for concrete beams of a certain type was introduced in Example 1.2. $$\begin{array} { l l l l l l l } 9.2 & 9.7 & 8.8 & 10.7 & 8.4 & 8.7 & 10.7 \\6.9 & 8.2 & 8.3 & 7.3 & 9.1 & 7.8 & 8.0 \\8.6 & 7.8 & 7.5 & 8.0 & 7.3 & 8.9 & 10.0 \\8.8 & 8.7 & 12.6 & 12.3 & 12.8 & 11.7 &\end{array}$$

A) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion, and state which estimator you used. Hint: $\sum x _ { i } = 246.8 .$
B) Calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%, and state which estimator you used.
C) Calculate and interpret a point estimate of the population standard deviation $\sigma$
Which estimator did you use? Hint: $\sum x _ { i } ^ { 2 } = 2327.54$
D) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 11 Mpa. Hint: Think of an observation as a "success" if it exceeds 11.
E) Calculate a point estimate of the population coefficient of variation $\sigma / \mu$
And state which estimator you used.

Question 26

Consider a random sample from the shifted exponential pdf
a. Obtain the maximum likelihood estimators of
b. A random sample of size
results in the values 3.12, .65, 2.56, 2.21, 5.45, 3.43, 10.40, 8.94, 17.83, and 1.31, calculate the estimates of

Question 27

Which of the following statements are true?

A) Maximizing the likelihood estimation is the most widely used estimation technique among statisticians.
B) Under very general conditions on the joint distribution of the sample, when the sample size n is large, the maximum likelihood estimator of any parameter $\theta$
Is approximately unbiased; that is, $E ( \hat { \theta } ) \approx \theta$
)
C) Under very general conditions on the joint distribution of the sample, when the sample size n is large, the maximum likelihood estimator of any parameter $\theta$
Has variance, is nearly as small as small as can be achieved by any estimator.
D) In recent years, statisticians have proposed an estimator, called an M-estimator, which is based on a generalization of maximum likelihood estimation.
E) All of the above are true statements.

Question 28

The shear strength of each of ten test spot welds is determined, yielding the following data (psi):
a. Assuming that shear strength is normally distributed, estimate the true average shear strength and standard deviation of shear strength using the method of maximum likelihood.
b. Again assuming a normal distribution, estimate the strength value below which 95% of all welds will have their strengths. (Hint: What is the 95 percentile in terms of
? Now use the invariance principle.)